Let \(\xi =\frac{y}{t}<0\) , then \(z_0=\sqrt{-\frac{1}{4\xi }}\in \mathbb {R}\) , where \(\pm z_0\)  are the two critical points of the phase function \(\theta (z)\) .
The case of \(\xi >0\)  is discussed by Xu [1]}.
Then (REF ) can be written as
\(\theta (z)=-\frac{z}{4}\left(\frac{1}{z_0^2} +\frac{1}{z^2} \right) ,\hspace{14.22636pt}\text{Re} (2it\theta )=-2t\text{Im}\theta =-2t\text{Im}z\left(\xi +\frac{1}{4|z|^2} \right) .\) 
